Lemma 13 (Lemma B.9, [1]}) 

Given a locally Lipschitz definable \(f:D\rightarrow \mathbb {R}\)  with an open domain \(D\) , for any interval \(I\)  and any arc \(z:I\rightarrow D\) , it holds for a.e. \(t\in I\)  that
\(\frac{\mathrm {d} f\left(z_{t}\right)}{\mathrm {d} t}=\left\langle z_{t}^{*}, \frac{\mathrm {d} z_{t}}{\mathrm {d} t}\right\rangle , \quad \text{ for all } z_{t}^{*} \in \partial f\left(z_{t}\right).\) 

